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Let $A$ be an algebra with finite dominant dimension $d \geq 1$ and $Dom_d$ the full subcategory of modules with dominant dimension at least $d$ and $Proj$ the full subcategory of modules of finite projective dimension. It is well known that the finitistic dimension conjecture for $A$ is true in case $Proj$ is contravariant finite. Now one can show that the finitstic dimension of $A$ is also finite in case the smaller subcategory $Dom_d \cap Proj$ is contravariant finite. Question:

-Is there an example where $Dom_d \cap Proj$ is contravariant finite, but $Proj$ is not? (One might try to choose $Dom_d$ to be representation-finite in a non-representation-finite algebra $A$)

-Is there an example where $Dom_d \cap Proj$ is not contravariant finite? (very likely yes. But if there is no such example, one could get the surprising result that the finitistic dimension conjecture is equivalent to the Nakayama conjecture for algebras with positive dominant dimension.)

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